Maths How To with Anita

# Sine 30 degrees (exact value, proof and example problems)

Sine 30 degrees is one of the most important trigonometric values. It has many applications in mathematics and physics.

This article will discuss what sine 30 degrees is, a geometrical proof of it’s value and how to find equivalent trig values.

We will also show a video demonstration of some example problems with solutions.

## What is the exact value of sin 30 degrees?

The value of sine 30 degrees is 1/2.

As a decimal you could say sin 30°=0.5

In radians, sin 30° = sin (π/6) = 1/2

## Derivation of sine 30 degrees (Geometrical method)

This can be proved using geometry. Draw an equilateral triangle of length 2cm. Each angle is 60°.

Construct a perpendicular bisector from one side to the opposite angle.

This bisects the 60° angle. So a 30° angle is formed along with a side of 1cm.

Since the trigonometric ratio is sine is equal to the opposite side/ hypotenuse.

Label the 1cm side opposite to the 30° angle as opposite and the 2cm side opposite the right angle as hypotenuse as follows:

So we can see sin 30 = 1/2

This is the exact value of sine 30 degrees.

## Trigonometric Ratios

There are 6 trigonometric ratios: sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec) and cotangent (cot).

Out of these, sine, cosine and tangent are the most important. They are defined as:

sin θ = opposite/hypotenuse

where the opposite side is opposite the angle of interest usally marked as θ or with an actual value.

The adjacent side is next to the angle of interest.

The hypotenuse is opposite the right angle. It is always the longest side in a right-angled triangle.

The other three trigonometric ratios can be derived from these.

They are reciprocal ratios:

cosecant (cosec) θ = 1/sin θ

secant (sec) θ = 1/cos θ

cotangent (cot) θ = 1/tan θ

## Trigonometry table

Finding the exact values of the common angles 0°, 30°, 45°, 60° and 90° is important for solving many problems.

This trigonometry table shows the exact values of these angles.

## Complementary trig values

Complementary angles are two angles whose sum is equal to 90°.

Finding equivalent trigonometric values is important for solving various trig problems.

In many problems, we are given only one value of a trigonometric ratio. But we need to find another value which is equal to it.

For example, if we are asked to find cos 60° we can say it equals 1/2 too because 60° + 30° = 90°.

So sin 30° = cos 60° = 1/2

## Sine 30 degrees on the Unit Circle

Sine 30 degrees can be found on the unit circle as it is the y co-ordinate of the point that is 30 degrees from the positive direction of the x axis.

As the y coordinate is 0.5, sin 30° = 0.5

## Why is sine 150 degrees equal to sin 30 degrees?

150° = 180°-30°

So sine 150 degress is equal to sine 30 degrees because 150 degrees is in the second quadrant where sine is positive and the related angle is 30 degrees.

## Equivalent values of sin 30

These are some other values which sine 30 can be written as:

sin 30° = sin (150 + 360n)°, where n is an integer

sin 30° = sin (30 + 360n)°, where n is an integer

sin 30° = 1/ cosec 30°

sin 30° = 1/sec 60°

## Example problems

Question: Is -sin 30° the same as sin (-30)°?

Solution: Yes, sin (-30)° is in the 4th quadrant. It is measured clockwise from 0°. Sine is negative in the 4th qudrant, so sin(-30)° = -sin 30° = 1/2

Question: Find the exact value of sin 210°.

Solution: 210° = (180 + 30)° so this is in the 3rd quadrant and 30° is the related angle. Sine is negative in the 3rd quadrant so:

sin 210° = – sin 30°

= – 1/2

Question: Find the exact value of cosec 330°.

Solution: 330° = (360 – 30)° so this is in the 4th quadrant and 30° is the related angle. Cosec is the reciprocal of sine and sine is negative in the 4th quadrant so cosec is also negative.

cosec 330° = 1/ sin 330°

= 1/-sin 30°

= 1/ – 1/2

= -2

Question: A ship sails 50 nautical miles due north from X to Y, then 25 nautical miles due east from Y to Z. Find θ, the bearing of Z from X, correct to the nearest degree.

Then using sin θ = opposite/ hypotenuse,

sin θ = 25/50

∴ sin θ = 1/2

∴ θ = 30°

## FAQs

### What is the sin and cos of 30 degrees?

Sin 30 degrees is 1/2 and cos 30 degrees equals 3/2.

### How do you find sin 30 degrees without a calculator?

Use the unit circle. sin 30 is the y co-ordinate of the point that is 30 degrees above the x axis. This is 0.5

### How do you do sin 30 on a calculator?

Press ‘sin’ then 30 and =

0.5 = 1/2

## Conclusion

Learning the exact value of sin 30 is 1/2 can be really helpful when trying to solve trigonometry problems. I hope this has helped and if you want to learn more about trignonometry, click here. Stay sharp!